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Non-Linear Time Series pp 199–244Cite as

Models for Integer-Valued Time Series

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Notes

  1. 1.

    Several other approaches have been proposed in the literature for the analysis of the time series of counts, including static regression models and autoregressive conditional mean models; see Jung and Tremayne (2011) for further details.

  2. 2.

    A r.v. Y (a) is said to be self-generalized with respect to parameter a, if \(P_{Y (a)}(P_{Y (a)}(s; a); a') = P_{Y (a)}(s; \mathit{aa}')\), for all a, a′ ∈ [0, 1].

  3. 3.

    An excellent account for the theory of INAR(1) models with a finite range of counts can be found in Weiß and Kim (2013) and the references therein.

  4. 4.

    The case a ∈ (0, 1) is called the stable case. Ispány et al. (2003) introduced a nearly-unstable INAR(1) model with a n  = 1 −δ n ∕n, and \(\delta _{n} \rightarrow \delta\) as n → ∞.

  5. 5.

    Although for the INAR(1) model the values of the ACF are always non-negative.

  6. 6.

    A discrete distribution in \(\mathbb{N}_{0}\) with probability generating function P(z) is called DSD if \(P(z) = P(1 - a + \mathit{az})P_{a}(z)\) , for |z| < 1 and a ∈ (0, 1), with P a (⋅) being some probability generating function. Alternatively, a non-negative integer-valued random variable X is DSD if for each a ∈ (0, 1) there is a non-negative random variable X a such that \(X\stackrel{d}{=}a \circ X' + X_{a}\) , where a ∘ X′ and X a independent, and X′ is distributed as X.

  7. 7.

    In what follows we will omit the index t below the thinning operator if there is no risk of misinterpretation.

  8. 8.

    Note that condition (5.29) is necessary and sufficient for the existence of such a sequence and of a real sequence (u n ) such that \(k_{n}(1 - F(u_{n})) \rightarrow \tau > 0\), as n → ∞.

  9. 9.

    Results for periodic sequences with exponential-type tails as in (5.30) can be found in Hall and Scotto (2006).

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Authors and Affiliations

  1. Departmento de Estatística e Investigação Operacional, Faculdade de Ciências Universidade de Lisboa, Lisboa, Portugal

    Kamil Feridun Turkman & Patrícia de Zea Bermudez

  2. Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal

    Manuel González Scotto

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Turkman, K.F., Scotto, M.G., de Zea Bermudez, P. (2014). Models for Integer-Valued Time Series. In: Non-Linear Time Series. Springer, Cham. https://doi.org/10.1007/978-3-319-07028-5_5

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